Using NeuralPDE.jl to solve differential equations

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This paper describes the application of physics-informed neural network (PINN) for solving partial derivative equations. Physics Informed Neural Network is a type of deep learning that takes into account physical laws to solve physical equations more efficiently compared to classical methods. The solution of partial derivative equations (PDEs) is of most interest, since numerical methods and classical deep learning methods are inefficient and too difficult to tune in cases when the complex physics of the process needs to be taken into account. The advantage of PINN is that it minimizes a loss function during training, which takes into account the constraints of the system and th e laws of the domain. In this paper, we consider the solution of ordinary differential equations (ODEs) and PDEs using PINN, and then compare the efficiency and accuracy of this solution method compared to classical methods. The solution is implemented in the Julia programming language. We use NeuralPDE.jl, a package containing methods for solving equations in partial derivatives using physics-based neural networks. The classical method for solving PDEs is implemented through the DifferentialEquations.jl library. As a result, a comparative analysis of the considered solution methods for ODEs and PDEs has been performed, and an evaluation of their performance and accuracy has been obtained. In this paper we have demonstrated the basic capabilities of the NeuralPDE.jl package and its efficiency in comparison with numerical methods.

作者简介

Daria Belicheva

RUDN University

Email: dari.belicheva@yandex.ru
ORCID iD: 0009-0007-0072-0453

Master student of Department of Probability Theory and Cyber Security

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Ekaterina Demidova

RUDN University

Email: eademid@gmail.com
ORCID iD: 0009-0005-2255-4025

Master student of Department of Probability Theory and Cyber Security

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Kristina Shtepa

RUDN University

Email: shtepa-ka@rudn.ru
ORCID iD: 0000-0002-4092-4326
Researcher ID: GLS-1445-2022

Assistent Professor of Department of Probability Theory and Cyber Security

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Migran Gevorkyan

RUDN University

Email: gevorkyan-mn@rudn.ru
ORCID iD: 0000-0002-4834-4895
Scopus 作者 ID: 57190004380
Researcher ID: E-9214-2016

Docent, Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Anna Korolkova

RUDN University

Email: korolkova-av@rudn.ru
ORCID iD: 0000-0001-7141-7610
Scopus 作者 ID: 36968057600
Researcher ID: I-3191-2013

Docent, Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Dmitry Kulyabov

RUDN University; Joint Institute for Nuclear Research

编辑信件的主要联系方式.
Email: kulyabov-ds@rudn.ru
ORCID iD: 0000-0002-0877-7063
Scopus 作者 ID: 35194130800
Researcher ID: I-3183-2013

Professor, Doctor of Sciences in Physics and Mathematics, Professor of Department of Probability Theory and Cyber Security of RUDN University; Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; 6 Joliot-Curie St, Dubna, 141980, Russian Federation

参考

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  20. Stepa, C. A., Fedorov, A. V., Gevorkyan, M. N., Korolkova, A. V. & Kulyabov, D. S. Solving the eikonal equation by the FSM method in Julia language. Discrete and Continuous Models and Applied Computational Science 32, 48-60. doi: 10.22363/2658-4670-2024-32-1-48-60 (2024).
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